Editing both 30fps and 60fps shots

Ok so I'm feeling like a novice here. I shot both 1080/30 and 720/60. Usually when I shot 60p I just edit in a 60p timeline and then export as a 29.97. For slow motion I can slow down 50% and it's not ideal but it works for the applications I've had in the past.

So I dropped my 30fps shots into a 60fps timeline that I'm editing in. I noticed that my 60p shots slowed down have a little blur between frames when I go frame by frame (see the edge of his hat in image below).

So I conformed the same clip to 24 and dropped it in. It obviously plays back as slow motion and looks better becuase I lost the wierd half frames. The only problem is when I speed it up (250%) to play at normal speed in my 60p timeline I'm seeing the half frames again. I can't seem to get away from this crap and it's making me feel slightly incompetent as an editor. Any help or insight would be greatly appreciated.

You have frame blending turned on. What application are you editing in?

I would recommend never editing at 60, since you won't be exporting at 60 (basically, always edit at the frame rate that you plan on exporting at). Because if you edit at 60fps, and you slow your 60p footage down by any amount you will either have duplicate frames or blended frames. The same with 30fps footage - every other frame will be duplicated. I'd recommend editing at 30fps, and interpreting your 60fps footage at 30. Then just speed up your 60fps footage up if you don't want it playing in slow mo.

I just tried dropping some 30p footage into a 60 timeline in Premiere CS6 and every other frame is a duplicate. It's not noticeable when it plays since it's no different than viewing at 30p, but there are definitely dupe frames.